Abstract:The theory of critical curves for maps of the plane provides powerful tools for locating the chief characteristic features of a discrete dynamical system in two dimensions: the location of its chaotic attractors, its basin boundaries, and the mechanisms of its bifurcations. Nowadays one begins to recognize the role played by critical curves of maps in the analysis, in the understanding and description of the bifurcations, and transition to chaotic behavior in coupled maps. In this paper we consider some properties of such maps, which possess a chaotic attractor. Some examples are considered in this paper in which we can see the effective role played by such curves in bifurcation theory.
Keywords: Homoclinic points, critical curves, bifurcations in endomorphisms, two-dimensional maps.